| Exam Board | CAIE |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2019 |
| Session | November |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Second order differential equations |
| Type | Solve via substitution then back-substitute |
| Difficulty | Challenging +1.8 This Further Maths question requires multiple sophisticated steps: deriving derivatives of w from y using chain rule, algebraic manipulation with trigonometric identities to verify the transformed DE, solving a second-order linear DE with particular integral, then back-substituting through inverse trig. The verification in part (i) demands careful algebraic skill, and converting initial conditions between variables adds complexity. Significantly harder than standard A-level but represents expected FM difficulty rather than exceptional challenge. |
| Spec | 4.10c Integrating factor: first order equations4.10e Second order non-homogeneous: complementary + particular integral |
It is given that $w = \cos y$ and
$$\tan y \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + \left( \frac { \mathrm { d } y } { \mathrm {~d} x } \right) ^ { 2 } + 2 \tan y \frac { \mathrm {~d} y } { \mathrm {~d} x } = 1 + \mathrm { e } ^ { - 2 x } \sec y$$
(i) Show that
$$\frac { \mathrm { d } ^ { 2 } w } { \mathrm {~d} x ^ { 2 } } + 2 \frac { \mathrm {~d} w } { \mathrm {~d} x } + w = - \mathrm { e } ^ { - 2 x }$$
(ii) Find the particular solution for $y$ in terms of $x$, given that when $x = 0 , y = \frac { 1 } { 3 } \pi$ and $\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { \sqrt { 3 } }$. [10]\\
\hfill \mbox{\textit{CAIE FP1 2019 Q11 EITHER [10]}}